# DETERMINING OPTIONS PREMIUM AND THE GREEKS

The Options premium is determined by three factors; namely, intrinsic value, time value and volatility. Now, there are more sophisticated tools applied to measure the potential variations of and in the Options premiums. These tools would be the Delta, the Gamma, the Theta, the Vega and the Rho. These have been explained in some detail below:

Delta: Delta is the measure of an Option's sensitivity to the changes in the price of the underlying asset. Therefore, it is the degree to which an Option price would move given a change in the price of the underlying asset or stock or index, while keeping all other factors as a constant.

Delta = (Change in Option premium) ÷ (Change in underlying price)

For instance, an Option with a Delta of 0.50 would move ₹5.00 for every change of ₹10.00 in the price of the underlying stock or index. To illustrate this further; let's say a trader is considering buying a Call option on a futures contract, which has a price of ₹19.00. The premium for the Call option with a strike price of ₹19.00 is ₹0.80. The Delta of the Option is +0.50. This means that, if the price of the underlying future contract rises to ₹20.00, which is a rise of ₹1.00; then the premium of the Option would increase by 0.50 multiplied with ₹1.0, which is 0.50. Thus, the new Option premium would be 0.80 + 0.50, which is ₹1.30.

Now, far out-of-the-money calls would have a very close to zero premiums, as the change in the underlying price is not likely to make them valuable or cheap. While, at-the-money calls would have a delta of 0.50 and a deeply in-the-money call would have a delta of 1.00. It may be noted here that, if the delta of our position is positive, then we would desire the price of the underlying asset to rise; and on the contrary, if the delta were to be negative we would want the price of the underlying to fall.

The knowledge and understanding of the Delta would be of vital importance to the Options traders as this parameter is frequently applied in margining and risk management strategies. The Delta is often called the hedge ratio. For instance, if we have a portfolio of 'n' shares of a stock, then 'n' divided by the Delta would give us the number of Call options we would need to be short (that is, would need to write) to create a riskless hedge; that is, a portfolio which would be worth the same monetary value whether the stock price rose or fell by a small amount. In such a scenario of a Delta neutral portfolio, any gain in the value of the shares held due to a rise in the stock price would be exactly offset by a loss in the value of the Calls written, and vice versa.

It may further be noted that, as the Delta changes with the stock price and time to expiration the number of shares would need to be continually adjusted to maintain the hedge. Now, how quickly the Delta changes with the stock price would be given by the Gamma.

Gamma: The Gamma is the rate at which the Delta value of an Option increases or decreases as a result of a move in the price of the underlying instrument (which may also be read as the underlying, or the asset or the stock, etc.).

Gamma = (Change in the Option Delta) ÷ (Change in the underlying price)

For instance, if a Call option has a Delta of +0.50 and a Gamma of 0.05, then a rise of +1 in the underlying means that the Delta will move to 0.55 for a price rise and 0.45 for a price fall. Gamma would be rather like the rate of change in the speed of an car (its acceleration) in moving from standstill upto its cruising speed and breaking back to a standstill. The Gamma would be the greatest for an at-the-money Option (cruising) and falling to zero as an Option moves deeply in-the-money and out-of-the-money (standstill).

If we were to be hedging a portfolio using the Delta hedge technique explained above, then we would want to keep the Gamma as small as possible; as the smaller it is, the less often we would be required to adjust the hedge to maintain the Delta neutral nature of the position. If the Gamma were to be too large, a small change in the stock price could wreck the hedge. Adjusting the Gamma can be tricky, and is usually done by buying or selling the underlying asset as the Gamma of the underlying asset is by definition always zero, so more or less of it would not affect the Gamma of the total portfolio.

Theta: The Theta is a measure of an Option's sensitivity to time decay; and would be the measure of the change in Option price given a one day decrease in time to expiration. It is a measure of time decay or time shrunk; and would give us an idea of how time decay would affect our portfolio.

Theta = (Change in Option premium) ÷ (Change in time to expiry)

The Theta is usually negative for an Option, as with a decrease in time the Option value decreases. This is due to the fact that the uncertainty element in the price decreases. Assume an Option has a premium of 3.00 and a Theta of 0.06, then after a day the premium would decrease to 2.94 and to 2.88 the following day. Naturally, other factors such as a change in value of the underlying stock would alter the premium; while the Theta is only concerned with the time value.

Unfortunately, we cannot predict with accuracy the changes in the stock's value or price, but we can measure exactly the time remaining until expiration.

Vega: The Vega is a measure of the sensitivity of an Option price to changes in market volatility; and would be the change of an Option premium for a given change (usually 1%) in the volatility of the underlying asset.

Vega = (Change in an Option premium) ÷ (Change in volatility)

For instance, if a stock has a volatility factor of 30%, the current premium is 3.00 with a Vega of 0.08; it would indicate that the premium would increase to 3.08 if the volatility factor increases by 1% to 30%. As the stock becomes more volatile the change in the premium would increase in the same proportion; as the Vega measures the sensitivity of the premium to these changes in volatility.

The Vega would be useful for option traders who while wishing to maintain a Delta neutral position would trade select Options purely in terms of their volatility; in such cases the trader would not be exposed to changes in the price of the underlying assets.

Rho: The Rho is the change in the Option price given a one percentage point change in the risk-free interest rate; and would measure the change in an Option's price per unit increase (usually 1%) in the cost of funding the underlying asset.

Rho = (Change in Option premium) ÷ (Change in cost of funding the underlying)

Assume the value of Rho is 14.10. Now, if the risk-free interest rate were to go up by 1%, the price of the Option would move by 0.14109. To put this in another way, if the risk-free interest rate changes by a small amount, then the Options value should change by 14.10 times that amount.

For example, if the risk-free interest rate increases by 0.01, from 10% to 11%, then the Option value would change by 14.10 multiplied with 0.01 which is 0.14. Now, for a Put option the relationship would be inverse. If the interest rate goes up the Option value decreases and therefore the Rho for a Put option is negative. In general Rho tends to be small, except in the case of long-dated Options.

We would strongly recommend that the investor first study these investment instruments along with the above listed factors affecting them; conduct dry runs with pen and paper; understand the nuances of the dynamics of the underlying security and the connectivity between the various risks that he or she would be taking on during the pendency of a futures or options position held by him.